OR_1_network I (hand..

Report
Network
Transportation
Assignment
Shortest Path
OPL
In Class
.
Lecture 09:
Network Programming I
.
Seeronk Prichanont1
Wipawee Tharmmaphornphilas1
Oran Kittithreerapronchai1
1
Department of Industrial Engineering, Chulalongkorn University
Bangkok 10330 THAILAND
last updated: February 10, 2014
OR I Network I – v1.0
1/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Outline
1.
2.
3.
4.
5.
6.
Importance of Network Programming
Transportation & Transhipment Problem
Assignment Problem
Shortest Path Problems
Solving Network Programming with OPL: Refinery
Problem
In-Class Exercise
OR I Network I – v1.0
2/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
What is Network Programming?
What: Network Programming is a set of LP problem that can represent by
Graph: Nodes and Arcs
Example: transhipment, shortest path, max-flow, min-spanning tree,
assignment
Why do we care about Network Problem?
Its applications are frequently emerged
Special structure that is easy to solve
Under ’right’ condition → integer solution for free
OR I Network I – v1.0
3/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Applications of Network Programming
Physical Networks
road networks
railway networks
airline traffic networks
electrical networks, e.g., the power grid
Abstract networks
project chart
multi-period inventory problem
OR I Network I – v1.0
4/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Introduction to Graph Theory
1
4
2.
3
Node: (vertex) presentation of stage
N ∈ {1, 2, 3, 4}
Edge: (arc) connecting line between nodes
E ∈ {(1, 2), (2, 3), (3, 4), (4, 1), (4, 3)}
OR I Network I – v1.0
5/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Parameters in Graph
-10
1
15
15, $2
4
-20
30, $2
8, $4
∞, $5
2.
10, $1
3
15
Demand/Suppy: demands and supply at each node, e.g. demands at node
1 is 10
Cost: incurred cost per a unit of flow, e.g., cost at edge (1, 2) is 5
Capacity: unit flow allow at each edge, e.g., flow at edge (1, 2) must be less
than ∞
OR I Network I – v1.0
6/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Example of Graph representation
A network with 6 vertices, N ∈ {1, 2, 3, 4, 5, 6}, and {−12, 24, −14, −16, 30, −12}
demands/suppies at vertices and the following cost and capacity at each edge. Construct
the graph of this network
Edge
(1,2)
(2,4)
(4,6)
(5,6)
(3,5)
(1,3)
(1,5)
(2,3)
(2,6)
(4,5)
OR I Network I – v1.0
Cost
$2
$2
$2
$2
$2
$2
$5
$1
$5
$1
Max Capacitiy
6
6
6
6
6
6
2
∞
2
∞
7/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Graph Representation
24
-16
2
4
-12
1
6
-12
3
5
-14
30
.
.
Convention signs
.
demand node = negative balance / inflow = negative coefficient
.
supply node = positive balance / outflow= positive coefficient
OR I Network I – v1.0
8/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Network Terminology
OR I Network I – v1.0
9/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Transportation Problems
Idea: Find the minimal total shipping cost with supply and demand limits
Observation: Two types of nodes: sources and destinations
Special algorithm: Transportation Simplex (Northwest corner)
Applications:
Minimize the total postal shipment costs
Minimize the total costs in production process
Related problems:
Bipartite Matching problem:
Transhipment problem: consisting of transit node (’0’ balance)
OR I Network I – v1.0
10/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
PowerCo Problem: Transmission Problem
Powerco has three power plants and must serve four cities with electricity
supplies/demands and transmission costs as follows:
To
From
Plant 1
Plant 2
Plant 3
Demand
(million kw-hr)
City 1
$8
$9
$14
45
City 2
$6
$12
$9
20
City 3
$10
$13
$16
30
City 4
$9
$7
$5
30
Supply
(million kw-hr)
35
50
40
Construct the graph representative of this network
source: Winston. 2003. Section 7.Ex01 pp. 360
OR I Network I – v1.0
11/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
PowerCo Problem: Network
35
50
40
-45
2
-20
3
-30
4
-30
2
3.
Plants
OR I Network I – v1.0
1
1
Cities
12/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Non-Balanced Transportation Problems
.
If
. total supplies ̸= total demands, introducing dummy node
In PowerCo problem, if plant 2 can generate only 40 million kw-hr (instead of 50 million
kw-hr).
35
1
1
-45
40
2
2
-20
40
3
3
-30
.
Plants
OR I Network I – v1.0
4 -30
Cities
13/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Boralis Problem: Production-Inventory
Problem
Boralis manufactures backpacks for serious hiker. Boralis estimates the demand for the
four months to be 100, 200, 180 and 300 units. Boralis can produce 50, 180, 280, and
270 units during month one to four. In addition,
The production cost per backpack is 40
The additional holding cost is $0.5 per backpack per month
The additional penalty cost is $2 incurred per backpack per month delay
Determine the optimal production schedule for the four months
source: Winston. 2003. Section 7.Ex01 pp. 328
OR I Network I – v1.0
14/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Boralis Problem: Network
50
-100
S1
D1
180
-200
S2
D2
280
S3
OR I Network I – v1.0
-180
.
D3
270
-300
S4
D4
source: production
dest: demand
15/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Widgetco Problem: Transhipment Problem
Widgetco manufactures at two factories, located in Memphis and Denver, and ships
widgets to two customer, located in LA and Boston. Each customer demands 130
widgets per day. The company can either ship directly to a customer or ship through its
two transit points: New York and Chicago. The transportation costs (in $ per widget)
are listed as follows:
From
Memphis
Denver
New York
Chicago
New York
8
15
6
To
Chicago
13
12
6
-
LA
25
26
16
14
Boston
28
25
17
16
If the Memphis and Denver factories have daily production of 150 and 200 widgets per
day.
source: Winston. 2003. Section 7.Ex05 pp. 400
OR I Network I – v1.0
16/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Widgetco Problem: Network
150
0
-130
ME
NY
LA
.
OR I Network I – v1.0
200
0
-130
DE
CH
BO
17/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Assignment Problems
What: A special application of transportation problems
Observation: all supplies equal 1; all demands equal 1
Decision{variable: ’0’ or ’1’
1,
if resouce i is assigned to object j
xi,j =
0,
otherwise.
Trivial: generalization Bipartite matching
OR I Network I – v1.0
18/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Bidding School Bus Problem
Four companies have made bids for school bus routes as follows:
Company
1
2
3
4
Route 1
$4000
$3000
-
Bids
Route 2 Route 3
$5000
$4000
$2000
$4000
Route 4
$5000
$5000
If each company can be assigned only one route, formulate assignment problem to
minimize cost of running the four bus routes
source: Winston. 2003. Section 7.5.8 pp. 400
OR I Network I – v1.0
19/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Bidding School Bus Problem: Graph
.
1
-1
C1
R1
1
-1
C2
R2
1
-1
C3
R3
1
-1
C4
R4
If each company can be assigned up to two routes, formulate assignment
problem to minimize cost of running the four bus routes
OR I Network I – v1.0
20/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Machine Scheduling Problem
source: Winston. 2003. Section 7.Ex04 pp. 400
OR I Network I – v1.0
21/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Machine Scheduling Problem: Graph
.
OR I Network I – v1.0
1
-1
M1
J1
1
-1
M2
J2
1
-1
M3
J3
1
-1
M4
J4
22/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Shortest Path Problems
Idea: Find a shortest path through the network that starts at an origin
(source) node to a destination (sink)node
Observation: Each edge has cost that incurs when used, but no capacity
Special algorithm: Dijkstra’s alorithm which can find the shortest paths
from one node to all others in a graph with non-negative arc lengths.
Applications:
Minimize the total distance traveled
Minimize the total cost of a sequence of activities
Minimize the total time of a sequence of activities
OR I Network I – v1.0
23/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Example of Shortest Path Problem
$3
2
4
$2
$4
Plant
$2
1
6
$3
City
$2
$3
3
5
.
OR I Network I – v1.0
24/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Library Shelving Problem
A library must build shelves to store 200 4-inch high books, 100 8-inch high books, and
80 12-inch high books. Each book is 0.5 inch thick. Each shelve costs $2,300 to build a
shelf and $5 per square inch is incurred to store (Assume that storage area = book’s
thickness × shelve hight). Formulate the problem as Shortest Path Problem.
Shelves
4”
8”
8”
8”
12”
12”
12”
Books
4”
8”
4”,8”
4”
12”
8”,12”
4”,8”,12”
Building Costs
$2300
$2300
$2300
$2300
$2300
$2300
$2300
Storage Costs
$2,000
$2,000
$6,000
$4,000
$2,400
$5,400
$11,400
Un-shelved
8”, 12”
4”, 12”
12”
8”, 12”
4”, 8”
4”
-
source: Winston. 2003. Section 8.2.8 pp. 419
OR I Network I – v1.0
25/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Library Shelving Problem: Network
$8.3
1
$7.7
0
$4.3
0
.
OR I Network I – v1.0
0
$4.3
4
-1
$4.7
8
12
$13.7
26/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
RentCar Problem
RentCar is developing a 4-year planning horizon replacement plan for its car fleet. At the
start of each year, RentCar must decide whether a car should be kept or replaced. A car
must be in service between 1 year to 3 years. The cost associated with the replacement
at each year are listed as follows:
Year acquired
year 1
year 2
year 3
year 4
Replacement cost
($1000) at year
1
2
3
4.0 5.4
9.8
4.3 6.2
8.7
4.8 7.1
4.9
-
Formulate RentCar problem as a shortest path problem.
OR I Network I – v1.0
27/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
RentCar Problem: Network
1
0
0
0
-1
Y1
Y2
Y3
Y4
Y5
.
OR I Network I – v1.0
28/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
OPL for Network Problem
Easy: Network constraints are the same (balancing or capacity)
Efficient: Graph may be sparse (many nodes, but few edges) → tuple
tuple: Custom Data Structure
Idea: tell OPL exact edges and its data used in model
What: custom data (edge, cost, max capacity, min capacity)
For: sparse graph
How:
input table contained all data (edge, cost, max capacity, min capacity)
extract set of edge, cost, max, and min of each edge
OR I Network I – v1.0
29/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Oilco Refinery: Transhipment Problem
Oilco has two oil fields (Los Angeles and San Diego), two uncapacitated refineries (Dallas and
Houston), and two customers (New York and Chicago). Each parties has demands, supply, and
transportation cost as follows.
City
Los Angeles
San Diego
Dallas
Houston
New York
Chicago
From
Los Angeles
San Diego
Dallas
Houston
Demand/Supply
Cost
(100,000 barrels) ($ per 100,000 barrels)
4
5
0
700
0
900
-3
-4
To
($ per 100,000 barrels)
Dallas Houston
New York Chicago
300
110
420
100
450
550
470
530
source: Winston. 2003. Section 8.5.5 pp. 455
OR I Network I – v1.0
30/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
OilCo Refinery: Network
4
0
-3
L
D
N
5
0
-4
S
H
C
-2
.
Field
OR I Network I – v1.0
dummy
Refinery
Customer
31/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
“tuple”: Tool for Sparse Matrix
Idea: list/create DVs and parameters only what we need
Example:
tuple DataTuple = {
string o;
string d;
float cost; };
{DataTuple} Data = ...;
tuple ConnectionTuple = {
string o;
string d; };
{ConnectionTuple} Connection = { < o,d > | < o,d,cost > in Data } ;
{float } transCosts[Connection] = [ < aData.o,aData.d >:aData.cost | aData in Data];
{float } demands[Connection] = ...;
OR I Network I – v1.0
32/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Oilco Refinery: Notation
Cities = {L,S,D,H,N,C ,dummy } ;
{string } Cities = ...;
z}|{
N
=
{<o,d>|<o,d,cost> in Data} ;
{ConnectionTuple} Connection =
z}|{
E
=
=
float operCosts[Cities] = ...;
z}|{
oi
etc.]#
z
}|
{
demands/supplies at node
;
[<aData.o,aData.d>:aData.cost|aData in Data] ;
float+ transCosts[Connection] =
z}|{
ci,j
z
}|
{
set of edges
demands = #[L:4, S:5, D:0,
float demands[Cities] =... ;
z}|{
Di
z
}|
{
set of vertices
=
z
}|
{
transportation costs at edge (i, j)
operCosts = #[D:700, H:900]# ;
z
}|
{
= operation cost at node i
dvar float+ flow [Connection] ;
z}|{
xi,j
OR I Network I – v1.0
= costs at edge (i, j)
33/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Oilco Refinery: Formulation
sum(<o,d> in Connection) (transCosts[<o,d>]+operCosts[o]) ∗ flow [<o,d>];
dexpr float objective
max
z}|{
z
}| {
z∑
=
z
}|
{
(ci,j + oi )xi,j
(i,j)∈ E
s.t.
sum(<o,c> in Connection)
z∑
}| {
(i,c)∈ E
OR I Network I – v1.0
sum(<c,d> in Connection)
flow [<o,c>]
z }| {
−xi,c
+
z∑
}| {
demands[c]
flow [<c,d>]
z}|{
xc,j
=
z}|{
Dc
forall(c in Cities)
z }| {
∀c∈ N
(c,j)∈ E
34/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Oilco Refinery: OPL model file
OR I Network I – v1.0
35/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Oilco Refinery: OPL data file
OR I Network I – v1.0
36/ 37
Network
Transportation
Assignment
Shortest Path
OPL
In Class
Job Assignment Problem
A company needs to assign 6 different jobs to 6 workers. Each worker has different skills
and may not be suitable for a particular job. Based on past experience, the supervisor
estimates the time for each worker to complete each job in minute as follow :
Worker/Job
A
B
C
D
E
F
1
9
8
-
2
15
10
3
10
13
-
4
14
16
14
5
11
15
13
-
6
10
6
Formulate mathematical programming to minimize total times
OR I Network I – v1.0
37/ 37

similar documents